On the sum of a prime and two Fibonacci numbers

Ji-Zhen Xu; Yong-Gao Chen

REVIEW 2 major objections 4 minor 22 references

On the sum of a prime and two Fibonacci numbers

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves three simultaneous facts about the representation function $r(n)$: an infinite arithmetic progression with no representations, a positive proportion with exactly one representation, and a positive proportion with at least…

desk verdict Genuine new result for sums of a prime and two Fibonacci numbers with square indices, but the second moment bound in Lemma 3.7 rests on an unproved uniqueness assertion that needs a real proof. read the letter →

arxiv 2506.03631 v1 pith:SVWAH4FK submitted 2025-06-04 math.NT

Ji-Zhen Xu , Yong-Gao Chen This is my paper

classification math.NT MSC 11P3211A4111B3911B13

keywords RomanofftypeproblemsprimesFibonaccinumbersapplicationsofsievemethodsasymptoticdensityrepresentationfunctionssquaresadditivebases

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies which integers can be written as a prime plus two Fibonacci numbers whose indices are perfect squares, $n=p+f_{k_1^2}+f_{k_2^2}$, and counts representations with $r(n)$. It proves that an entire arithmetic progression of positive integers has $r(n)=0$, that a positive proportion of integers have $r(n)=1$, and that a positive proportion have $r(n)\ge 2$. The interest is that the Fibonacci-square subsequence is very sparse, yet together with the primes it still covers a positive proportion of integers in a way that is both unique and multiple on positive-density sets. The proofs combine a congruence obstruction built from Fibonacci periodicity with sieve estimates that control collisions between different representations.

What carries the argument

The central object is a large modulus $M=621386267972593776074029725204132260351094$, the product of sixteen primes, together with a residue $n_0\pmod M$ and a prime residue $p_0\pmod M$ determined by the Chinese remainder theorem. The mechanism is Fibonacci periodicity: for each prime $q\mid M$, the Fibonacci sequence modulo $q$ has period $u(q)$, the least positive $u$ with $f_u\equiv f_0$ and $f_{u+1}\equiv f_1\pmod q$, and $v(d)=\max_{p\mid d}u(p)$ measures how far a modulus $d$ can detect a difference $f_{k_1^2}-f_{h_1^2}$. The proof forces $n\equiv n_0\pmod M$ to imply $k_i\equiv0,64,128\pmod{192}$ and $p\equiv p_0\pmod M$, then uses an upper-bound sieve for prime pairs to show that collisions $f_{k_1^2}+f_{k_2^2}=f_{\ell_1^2}+f_{\ell_2^2}$ are rare. The numerical estimate $\sum_{d\ge1}\mu^2(d)/(d\sqrt{[192^2,v(d)]})<0.23219$, where $[\cdot,\cdot]$ denotes the least common multiple, is what makes the second moment smaller than the first, producing positive density.

What would settle it

Take the unique residue $n_1\pmod{769M}$ with $n_1\equiv n_0\pmod M$ and $n_1\equiv501\pmod{769}$, and compute $r(n_1+769Mk)$ for the first few $k$; a single value above zero would disprove Theorem 1.1. A separate check is to enumerate $k_i,\ell_i\le10^6$ in the two index classes and test whether $f_{k_1^2}+f_{k_2^2}=f_{\ell_1^2}+f_{\ell_2^2}$ occurs with distinct pairs, which would invalidate the collision assertion used in Lemma 3.7.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the sparse set $\{f_{k^2}:k\in\mathbb N_0\}$ behaves like a two-term additive basis for the primes in a strong density sense. The paper constructs a modulus $M$ and a residue class $n_0\pmod M$ with the property that any representation of such an $n$ forces the prime to lie in a single residue class and both square indices to lie in two prescribed residue classes modulo $192$; intersecting that class with a further modulus $769$ produces an arithmetic progression with no representations at all. Averaging $r(n)$ over the residue class gives a positive first moment asymptotic to a constant times $x$, while a sieve bound on coincidences gives a second moment of order $x$; because the second-moment constant is small enough, both $\{n:r(n)=1\}$ and $\{n:r(n)\ge2\}$ acquire positive asymptotic density within the class, hence in the integers.

Load-bearing premise

The argument needs the fact that two different pairs of square-indexed Fibonacci numbers, with indices chosen from the special allowed sets, never add to the same total; the paper supports this only by citing the recurrence $f_m=f_{m-1}+f_{m-2}$, which by itself allows collisions such as $F_n+F_{n-3}=2F_{n-1}$.

Editorial extensions

If this is right

An infinite arithmetic progression of positive integers is missed entirely, so $\{p+f_{k_1^2}+f_{k_2^2}\}$ is not an additive basis in the strong sense of eventually representing all integers.
A positive proportion of all positive integers are represented exactly once, and a positive proportion are represented at least twice; both layers of the representation function are visible at the level of asymptotic density.
For the residue class $n\equiv n_0\pmod M$ the average count is asymptotic to $x/(2^{11}3^2\varphi(M)\log\alpha)$, and the second-moment bound implies that the set with $r(n)\ge T$ has density $O(1/T^2)$.
The boundary case of two square Fibonacci indices, where the reciprocals of the exponents sum to $1$, now has the finer description of unique and multiple representations alongside a missing progression, while the earlier result only gave positive proportion of represented integers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same CRT-plus-periodicity construction should transfer to $p+f_{k_1^s}+f_{k_2^s}$ for other exponents $s$: one needs a modulus whose Fibonacci periods divide a common multiple of the residues of $k^s$, and the index classes would force the same obstruction; the square case is one instance of a general template.
The numerical threshold in Lemma 3.6 is compared with $9\log\alpha$ at the end of the proof of Theorem 1.2, so the lower density of unique representations could be increased by sharpening either the sieve constant in Lemma 3.5 or the divisor-sum estimate; the paper's constants are not optimal.
Problem 1.5, asking whether every multiplicity $m$ occurs with positive proportion, remains open for $m\ge3$; the present proof only controls the second moment, so higher multiplicities would require higher moments or a different argument.
The method's dependence on Fibonacci periodicity suggests the same density phenomenon should hold for other Lucas sequences; the decisive quantity is whether the associated $v(d)$ satisfies a divisor-sum bound like Lemma 3.6.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline: this is a genuine new result in the Romanoff-type line, with a real gap in the proof of the second moment bound. The main construction—Theorem 2.1's congruence covering—is careful and explicit, and Theorems 1.1–1.3 would follow cleanly if Lemma 3.7 held as stated.

What is new: no one has treated p plus two Fibonacci numbers with square indices. The prior work on p+F_k and p+2^{a^2}+2^{b^2} does not make this routine; the rank-of-apparition covering needs new residue computations, and the authors supply them. The density results for r(n)=1 and r(n)≥2 are natural analogues of the Chen–Xu conjecture and worth having.

The soft spot is exactly what the stress-test flags. In Lemma 3.7, the assertion that f_{k_1^2}+f_{k_2^2}=f_{l_1^2}+f_{l_2^2} with k_i,l_i in K_i forces k_i=l_i is justified by one sentence citing the recurrence f_m=f_{m-1}+f_{m-2}. That is not a proof. The recurrence alone permits two-term sum collisions (e.g., F_n+F_{n+3}=F_{n+2}+F_{n+2}), and nothing in the text rules out such identities when the indices are squares in those congruence classes. This assertion is load-bearing: it is what lets the proof drop the p1=p2 terms in r(n)(r(n)-1). If the assertion fails, the second moment gets an additional term that is not controlled, and the positive density in Theorem 1.2 no longer follows from the written argument.

I don't think the assertion is obviously false; the square-index restriction plus the congruence classes may well eliminate all collisions. But the authors need to prove it, e.g., by citing a classification of F_x+F_y=F_z+F_w or by a valuation argument. As written, the gap is nontrivial and central.

Minor point: the large finite residue computations in Theorem 2.1 are stated but not carried out. That is acceptable at this level because they are explicit and checkable, but a referee should ask for a verification script or a reproducibility note.

Who this is for: people working on Romanoff-type problems and additive bases with sparse sequences. The paper deserves a serious referee; the gap is likely repairable, but the referee should push on Lemma 3.7 before acceptance.

Recommendation: send to peer review, and make the handling editor aware that the proof of Lemma 3.7 needs a substantive addition.

Referee Report

2 major / 4 minor

Summary. The paper studies the representation function r(n) counting representations of a positive integer n as p + f_{k_1^2} + f_{k_2^2}, where p is prime and k_1, k_2 are nonnegative integers with k_1 ≤ k_2. The main results are: Theorem 1.1 gives an infinite arithmetic progression of n with r(n)=0; Theorem 1.2 shows that a positive proportion of integers are represented uniquely; Theorem 1.3 shows that a positive proportion are represented in at least two ways. The proofs are built on a finite congruence covering (Theorem 2.1) that characterizes the allowed indices and primes for n ≡ n_0 (mod M), followed by first and second moment estimates using the prime number theorem in arithmetic progressions and sieve bounds.

Significance. If the proofs are correct, the results are a noteworthy analogue of classical Romanoff-type theorems: despite the extreme sparsity of the sequence {F_{n^2}}, it still behaves like an additive basis in a density sense. The paper is clearly organized, supplies explicit constants and moduli, and the main technical achievement is a second-moment estimate that is both elaborate and mostly explicit. The numerical bounds, such as Lemma 3.6, are derived in detail and appear plausible. The overall strategy is sound, provided the unresolved uniqueness issue in Lemma 3.7 and the related issue in Lemma 4.2 are fixed.

major comments (2)

[Section 3, Lemma 3.7] The proof of Lemma 3.7 opens with the assertion that if f_{k_1^2}+f_{k_2^2}=f_{l_1^2}+f_{l_2^2} with k_i,l_i ∈ K_i, then k_i = l_i, justified only by 'by f_m=f_{m-1}+f_{m-2} for all m≥2'. This is not a proof: the Fibonacci recurrence alone permits non-trivial equalities such as F_{n-1}+F_{n+2}=F_{n+1}+F_{n+1} and F_{n+1}+F_{n+2}=F_0+F_{n+3}. The restrictions that the indices are squares in the residue classes K_i may well rule out such collisions, but the paper supplies no argument to that effect. This assertion is load-bearing: it is exactly what allows the reduction of r(n)(r(n)-1) to the case p_1 ≠ p_2. If the assertion fails, the bound in Lemma 3.7, and consequently the positive density in Theorem 1.2, would not follow. A rigorous proof is needed, for instance a classification of all solutions to F_x+F_y=F_z+F_w with x,y,z,w in the relevant sets, or a valuation/residue argument exploiting that the indices are squares.
[Section 4, Lemma 4.2] In the proof of Lemma 4.2, the contribution to ∑_{n≤x} r(n)^2 from tuples with p_1 = p_2 is bounded by O(x) without justification. This term counts, for each prime p, pairs of index pairs (k_1,k_2) and (l_1,l_2) with f_{k_1^2}+f_{k_2^2}=f_{l_1^2}+f_{l_2^2}. Since the number of square-index pairs with sum ≤ x is O(log x), an a priori bound of O(x log x) is trivial; obtaining O(x) requires an upper bound on the number of representations of an integer as a sum of two Fibonacci numbers with square indices. No such bound is proved or cited. This is essentially the same uniqueness/collision question as in Lemma 3.7, and it is load-bearing for the estimate ∑ r(n)^2 ≪ x needed in the proof of Theorem 1.3. The authors should supply an argument that the number of non-diagonal collisions is O(x), or otherwise adjust the proof.

minor comments (4)

[Section 2, Proof of Theorem 1.1] The statement u(769)=192 is used without proof or verification; please add a short justification (e.g., a computation of F_{192} mod 769 and F_{193} mod 769) or a citation.
[Section 3, Lemma 3.6] The expression '5/2 log log(f_1...f_n)' is ambiguous; it should be written as 5/(2 log log(f_1...f_n)) to match the bound from [17, (3.41)].
[Abstract] The abstract says 'positive asymptotic densities', but the proofs establish positive lower density (positive proportion); consider rephrasing to 'positive lower density' or 'positive proportion' for precision.
[Section 3, Lemma 3.7] The notation P(d) for the largest prime divisor of d is used before it is defined; please define it at the point of first use.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the paper's constants are selected by explicit congruence conditions and the main estimates are cited to external sieve and prime-number theorems, not to its own conclusions.

full rationale

The derivation chain is self-contained rather than circular. Theorem 2.1 is proved by a finite case analysis using the Chinese remainder theorem and the periods u(q) of the Fibonacci sequence modulo each prime divisor of M; the large constants M, n0, and p0 are constructed residue classes, not fitted parameters. Lemma 3.2 uses the prime number theorem in arithmetic progressions and simple counting. Lemma 3.7 applies the Halberstam-Richert sieve bound (Lemma 3.5), the external Schinzel/Somer bound |B_d| <= 4, and an explicitly computed numerical estimate (Lemma 3.6). No step fits a parameter to the target quantity and then calls the consequence a prediction. The self-citations (e.g., Chen-Ding for limsup r(n)/loglog n) are background observations and are not load-bearing for Theorems 1.1 through 1.3. One non-circular concern should be flagged separately: Lemma 3.7 asserts, with only the one-line justification 'by f_m = f_{m-1} + f_{m-2} for all m>=2', that equality of two sums of Fibonacci squares with indices in K1, K2 forces equality of the index pairs. The recurrence alone does not make this obvious, and this injectivity claim is load-bearing for the second-moment bound. That is a correctness or proof-gap issue, not a circularity: the claim does not assume the theorem it is used to prove, and the surrounding estimates do not reduce to the target density statements.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard analytic number theory tools and on finite congruence computations for Fibonacci rank of apparition. There are no fitted parameters or invented entities; the modulus M and residue classes are explicit proof constructions rather than data-derived constants.

assumptions (5)

standard math Prime number theorem in arithmetic progressions
Used in Lemma 3.2 and Lemma 4.1 to estimate the count of primes in the residue class p0 mod M.
standard math Halberstam-Richert uniform upper-bound sieve for primes p and p+m in arithmetic progressions
Lemma 3.4 is the external sieve estimate used to bound the second moment in Lemma 3.7 and Lemma 4.2.
domain assumption Rank-of-apparition values u(q) for the 16 primes dividing M, as listed in (2.3)
The finite congruences in Theorem 2.1 depend on these exact values; the paper says 'A calculation shows'.
standard math Schinzel and Somer bound that f_x congruent to a mod p has at most 4 solutions in a period
Used in Lemma 3.7 to control the number of solutions k1^2 modulo [192^2, v(d)].
standard math Rosser-Schoenfeld inequality k/phi(k) <= e^gamma log log k + 5/(2 log log k) for k >= 3
Used in Lemma 3.6 to bound sums over squarefree divisors of products of Fibonacci numbers.

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Cite this review

Pith. "Pith review of On the sum of a prime and two Fibonacci numbers." pith.science (2026). https://pith.science/paper/SVWAH4FK

@misc{pith2026250603631,
  author       = {Pith},
  title        = {Pith review of: On the sum of a prime and two Fibonacci numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SVWAH4FK}},
  note         = {Machine review of arXiv:2506.03631}
}

abstract

Let $\{f_n\}$ be the Fibonacci sequence. For any positive integer $n$, let $r(n)$ be the number of solutions of $n=p+f_{k_1^{2}} +f_{k_{2}^{2}}$, where $p$ is a prime and $k_1, k_2$ are nonnegative integers with $k_1\le k_2$. In this paper, it is proved that $\{ n : r(n)=0\} $ contains an infinite arithmetic progression, and both sets $\{ n : r(n)=1\} $ and $\{ n : r(n)\ge 2\}$ have positive asymptotic densities.

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Apostol, Introduction to analytic number theory, Springer- Verlag, 1976

Tom M. Apostol, Introduction to analytic number theory, Springer- Verlag, 1976

work page 1976
[2]

Y. -G. Chen and Y. Ding,On a conjecture of Erd˝ os, C. R. Math. Acad. Sci. Paris 360 (2022), 971–974

work page 2022
[3]

Chen and Y

Y.-G. Chen and Y. Ding,Quantitative results of the Romanov type representation functions, Quart. J. Math. 74 (4) (2023), 1331–1359

work page 2023
[4]

Y. -G. Chen and R. -J. Wang,The sum of a prime and a term of exponential sequences(Chinese), Acta Math. Sinica (Chinese Ser.) 67 (2024), 259–272

work page 2024
[5]

Y. -G. Chen and J. -Z. Xu,On integers of the formp+ 2 kr1 1 +· · ·+ 2krt t , J. Number Theory 258 (2024), 66–93

work page 2024
[6]

Crocker,On the sum of a prime and two powers of two, Pacific J

R. Crocker,On the sum of a prime and two powers of two, Pacific J. Math. 36 (1971), 103–107

work page 1971
[7]

Davenport, Multiplicative Number Theory, Second edition, GTM 74, Springer-Verlag, New York, 1980

H. Davenport, Multiplicative Number Theory, Second edition, GTM 74, Springer-Verlag, New York, 1980

work page 1980
[8]

Ding,On a problem of Romanoff type, Acta Arith

Y. Ding,On a problem of Romanoff type, Acta Arith. 205 (2022), 53– 62

work page 2022

Show all 22 references

[9]

Elsholtz and J

C. Elsholtz and J. -C. Schlage-Puchta,On Romanov’s constant, Math. Z. 288 (2018), 713–724

2018
[10]

Erd˝ os,On integers of the form2 k +pand some related problems, Summa Brasil

P. Erd˝ os,On integers of the form2 k +pand some related problems, Summa Brasil. Math. 2 (1950), 113–123

1950
[11]

Halberstam and H

H. Halberstam and H. -E. Richert, Sieve Methods, Academic Press, London, 1974

1974
[12]

K. S. E. Lee,On the sum of a prime and a Fibonacci number, Int. J. Number Theory 6 (2010), 1669–1676

2010
[13]

Liu and M

Z. Liu and M. Xue,The sum of a prime and a Fibonacci number, Int. J. Number Theory 17 (2021), 1815–1823

2021
[14]

Pan,On the integers not of the formp+ 2 a + 2b, Acta Arith

H. Pan,On the integers not of the formp+ 2 a + 2b, Acta Arith. 148 (2011), 55–61. On integers of the formp+fk2 1 +f k2 2 21

2011
[15]

Pintz,A note on Romanoff ’s constant, Acta Math

J. Pintz,A note on Romanoff ’s constant, Acta Math. Hungar. 112 (2006), 1–14

2006
[16]

N. P. Romanoff, ¨Uber einige S¨ atze der additiven Zahlentheorie, Math. Ann. 109 (1934), 668–678

1934
[17]

J. B. Rosser and L. Schoenfeld,Approximate formulas for some func- tions of prime numbers, Illinois J. Math. 6 (1962), 64–94

1962
[18]

Schinzel,Special Lucas sequences, including the Fibonacci sequence, modulo a prime, inA Tribute to Paul Erd˝ os, eds

A. Schinzel,Special Lucas sequences, including the Fibonacci sequence, modulo a prime, inA Tribute to Paul Erd˝ os, eds. A. Baker, B. Bollobas and A. Hajnal (Cambridge University Press, Cambridge, 1990) 349– 357

1990
[19]

ˇSiurys,On integers not of the formF n ±p a, Int

J. ˇSiurys,On integers not of the formF n ±p a, Int. J. Number Theory 12 (2016), 505–512

2016
[20]

Somer,Distribution of residues of certain second-order linear re- currences modulop, inApplications of Fibonacci Numbers, eds

L. Somer,Distribution of residues of certain second-order linear re- currences modulop, inApplications of Fibonacci Numbers, eds. G. E. Berum, A. N. Philippou and A. F. Horadam, Vol. 3 (Kluwer Acad. Publ., 1990) 311–324

1990
[21]

R. -J. Wang and Y. -G. Chen,On the sum of a Fibonacci number and a prime, Int. J. Number Theory 19 (2023), 873–889

2023
[22]

J. -Z. Xu and Y. -G. Chen,On integers of the formp+ 2 a2 + 2b2 , Int. J. Number theory 21 (2025), 153–175

2025

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Apostol, Introduction to analytic number theory, Springer- Verlag, 1976

Tom M. Apostol, Introduction to analytic number theory, Springer- Verlag, 1976

work page 1976

[2] [2]

Y. -G. Chen and Y. Ding,On a conjecture of Erd˝ os, C. R. Math. Acad. Sci. Paris 360 (2022), 971–974

work page 2022

[3] [3]

Chen and Y

Y.-G. Chen and Y. Ding,Quantitative results of the Romanov type representation functions, Quart. J. Math. 74 (4) (2023), 1331–1359

work page 2023

[4] [4]

Y. -G. Chen and R. -J. Wang,The sum of a prime and a term of exponential sequences(Chinese), Acta Math. Sinica (Chinese Ser.) 67 (2024), 259–272

work page 2024

[5] [5]

Y. -G. Chen and J. -Z. Xu,On integers of the formp+ 2 kr1 1 +· · ·+ 2krt t , J. Number Theory 258 (2024), 66–93

work page 2024

[6] [6]

Crocker,On the sum of a prime and two powers of two, Pacific J

R. Crocker,On the sum of a prime and two powers of two, Pacific J. Math. 36 (1971), 103–107

work page 1971

[7] [7]

Davenport, Multiplicative Number Theory, Second edition, GTM 74, Springer-Verlag, New York, 1980

H. Davenport, Multiplicative Number Theory, Second edition, GTM 74, Springer-Verlag, New York, 1980

work page 1980

[8] [8]

Ding,On a problem of Romanoff type, Acta Arith

Y. Ding,On a problem of Romanoff type, Acta Arith. 205 (2022), 53– 62

work page 2022